synchronizing email between devices

You need to configure the mail accounts as IMAP. With an IMAP account the master copy of all messages remains on the server, and any time you read, move or delete a message on one platform that action will be replicated on the server, then to each of the mail clients.

Synchronized folders would be the ones on the Server & on Your Mac.

You can have a look on you Mail Prefs then check your account's Mailbox Behaviors and then see if the Trash is pointing to "Trash On My Mac" or "Trash on the Server". By selecting "Trash on the Server", this would delete the email in all devices since it is saved in the server.